To mention but one example, some of the above operators operators appear automatically in theories with non–minimal couplings of scalars to gravity [1]}.
In particular,
\((\partial _\mu s)^2 \phi ^2\) , \((\partial _\mu \phi )^2 s^2\)  and \({s^2 } \, V(\phi )\)  are induced already at tree level by
the metric transformation from the Jordan frame to the Einstein frame [2]}. The (unsuppressed) operators \(\phi ^2 s^4\)  and \(\phi ^4 s^2\) 
appear in these models
at 1–loop via the graviton loop. Their coefficients are proportional to the product of the non–minimal couplings and the loop factor, and thus expected to be significant (in the absence of
symmetry arguments).
In general, it is a challenging task to estimate
the relative size of the different operators since this can only be done reliably within
UV complete gravity theories.
